Spectral Graph Theory
Speaker: Gordon Royle
Abstract: This course is an introduction to spectral graph theory. The topics to be covered are:
- Linear Algebra, Adjacency Matrix, Laplacian Matrix, the Perron-Frobenius Theorem
- Adjacency spectra, spectra of graph families, spectral radius, interlacing, Hoffman bound, graphs with least eigenvalue -2
- Laplacian spectrum, algebraic connectivity, isoperimetric number
- Spectral decomposition, strongly regular and distance regular graphs, Moore graphs
- Open problems, computational exploration, spectral gap sets
Graphs and Coherent Algebras
Speaker: Gary Greaves
Abstract: This course will introduce coherent configurations and their application to studying certain families of graphs. One important motivation for the study of coherent configurations stems from the classical Graph Isomorphism Problem and the Weisfeiler-Leman stabilisation algorithm. Special cases of coherent configurations include Schur partitions, association schemes, and distance regular graphs, which have received intense study. This course will cover the basic theory of these objects and show how the coherent closure of graphs has been used recently to study certain families of graphs.
Variation on Antimagic Labeling
Speaker: Kiki Ariyanti Sugeng
Abstract: A graph with size is said to be antimagic if the edges can be labeled with the integers in such a way that the sum of the edge labels at any given vertex is unique, i.e., no two vertices have the same sum. In this lecture, I will share my experience conducting antimagic labeling research, including several variations of this labeling., such as edge antimagic total, vertex antimagic total and distance antimagic total labeling. The course will be divided into 3 lectures:
1. Edge antimagic total labeling
2. Vertex antimagic total labeling
3. Distance antimagic labeling.
Some Classical Results on Graph Coloring
Speaker: Reginaldo Marcelo
Abstract: Given a graph G = (V,E) and a set C of `colors’, a function c : V → C (resp. c : E → C) is called a proper vertex (resp. edge) coloring of G if no two adjacent vertices (resp. adjacent edges) are assigned the same color. Among all proper vertex colorings c of G, the minimum number of colors used is called the chromatic number of G and is denoted by χ(G). Similarly, the chromatic index of G, denoted by χ’(G) is the minimum number of colors used in a proper edge coloring of G. In this talk we discuss some classical results on the chromatic number and chromatic index of graphs. On chromatic numbers, we prove the Five Color Theorem and the Brook’s Theorem. We also review the history of the Four Color Theorem, culminating in the computer-assisted proof of Appel and Haken in 1976—marking its 50th anniversary in 2026. Concerning chromatic indices, we prove the Vizing’s Theorem and the Kӧnig’s Line-Coloring Theorem.
Analytic Combinatorics
Speaker: Michael Fuchs
Abstract: Analytic combinatorics is a relatively recent sub-branch of combinatorics. The purpose of this course is to give an introduction into this exciting area. We will start by explaining methods from symbolic combinatorics, which are used to build generating functions in a systematic way. Then, we will show how complex-analytic tools can be used to obtain asymptotic information about the sequences encoded by the generating functions. In the final part, we will give applications, in particular very recent ones to phylogenetic trees and networks which are fundamental graph-theoretical models in phylogenetics.
Graph Symmetry
Speakers: Binzhou Xia and Maria Carmen Amarra
Abstract: This course will cover the following topics:
- Graph symmetry properties
- Group-theoretic approach
- Cayley graphs
- Symmetry of graph products